Embedding finite groups in rational division algebras. I

Abstract

In [7] and [8] the problem of determining the finite subgroups of division rings with specified centers was investigated. In particular, we were concerned with determ~ng for which fields K every odd order finite subgroup of a finite-dimensional division ring central over K is necessarily cyclic. This problem was completely settled for K a local field in [8]. In this paper we answer this question when K is an algebraic number field. Our main result is the following: Let K be an algebraic number field. Then there exists a division ring D finite dimensional and central over K such that D* has a noncyclic subgroup of odd order if and only if K contains a primitive q-th root of unity for some odd prime q. We maintain the notation and terminology of [7J and [8]. Recall that for K a field a K-division ring D is a finite-dimensional central division algebra over K. The dimension of D over K will be denoted by ED: fcl; we use the same notation for dimension of field extensions. We say that D is E-adequate if there is an E-division ring containing I). If a finite group G is contained in the multiplicative group of a K-division ring, we say that G is K-adequate. G is a K-adequate if and only if V(G) is K-adequate where V(G) is the minimal division ring containing G; the structure of T(G) was determined by Amitsur in [2]. By an A-group we will mean a noncyclic odd order group which is a subgroup of some division ring.